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Limit

One of the fundamental concepts in mathematics, meaning that a variable depending on another variable arbitrary closely approaches some constant as the latter variable changes in a definite manner. In the definition of a limit, the concept of nearness of the objects under consideration plays a fundamental role: only after a definition of nearness does a limit acquire an exact meaning. The following fundamental concepts of mathematical analysis are connected with that of a limit: continuity, convergence, derivative, differential, integral. One of the simplest cases of a limit is the limit of a sequence.

The limit of a sequence.

Let $X$ be a topological space. A sequence of points $x_n$, $n=1,2,\dots,$ of $X$ is said to converge to a point $x_0\in X$ or, which is the same, the point $x_0$ is said to be a limit of the given sequence if for each neighbourhood $U$ of $x_0$ there is a natural number $N$ such that for all $n>N$ the membership $x_n\in U$ is satisfied. In this case one writes
$$
\lim_{n\to \infty} x_n = x_0.
$$
In the case when $X$ is a Hausdorff space, the limit of a sequence $x_n\in X$, $n=1,2,\dots,$ is unique, provided that it exists. For a metric space $X$, a point $x_0$ is the limit of a sequence $\{x_n\}$ if and only if for each $\epsilon >0$ there is a natural number $N$ such that for all indices $n>N$ the inequality $\rho(x_n,x_0)<\epsilon$ is satisfied, where $\rho(x,x_0)$ is the distance between $x_n$ and $x_0$. If a sequence of points of a metric space is convergent, then it is bounded. A sequence of points of a complete metric space is convergent if and only if it is a Cauchy sequence. In particular, this is true for sequences of numbers, for which the concept of a limit of a sequence historically arose first (see Cauchy criteria). For such sequences the following formulas hold:
$$
\lim_{n\to \infty} c = c, \quad
\lim_{n\to \infty} cx_n = c \lim_{n\to \infty} x_n,
$$
where $c$ is any given number;
$$
\lim_{n\to \infty} (x_n+y_n) = \lim_{n\to \infty} x_n + \lim_{n\to\infty}y_n;\\
\lim_{n\to \infty} x_n y_n = \lim_{n\to \infty} x_n \lim_{n\to \infty} y_n;
$$
and if $\lim_{n\to \infty} y_n \neq 0$, then
$$
\lim_{n\to\infty}\frac{x_n}{y_n} = \frac{\lim_{n\to \infty} x_n}{\lim_{n\to\infty} y_n}.
$$
These properties of sequences of numbers can be carried over to limits of sequences in more general structures, for example, the property of the limit of a sum — to sequences of points in linear topological spaces, the property of the limit of a product — to sequences of points in a topological group, etc.

Every increasing (decreasing) sequence of real numbers $x_n$, i.e. such that $x_n \le x_{n+1}$ ($x_n \ge x_{n+1}$), $n=1,2,\dots,$ that is bounded from above (below) is convergent and its limit is the supremum (infimum) of the set of its members. For example, if $\alpha>0$, $k$ is a natural number and $\alpha_n$ is an approximate value of the root $\alpha^{1/k}$ calculated up to $n$ decimal places after the decimal point, then the $\alpha_n$, $n=1,2,\dots,$ form an increasing sequence and $\lim_{n\to\infty}\alpha_n = \alpha^{1/k}$. Another example of an increasing sequence which is bounded from above is the sequence of perimeters of regular polygons with $n$ sides, $n=3,4,\dots,$ inscribed in some circle; this sequence converges to the length of the circle.

In the theory of sequences of numbers a fundamental role is played by infinitesimal sequences or null sequences, i.e. those sequences which converge to zero. The general concept of a sequence of numbers can be reduced to that of an infinitesimal sequence in the sense that a sequence of numbers converges to a given number if and only if the differences between the terms of the sequence and the given number form an infinitesimal sequence.

The concept of an infinitely-large sequence of numbers is also useful. These are the sequences with as limits one of the infinities $+\infty$, $-\infty$, or the infinity $\infty$ without a sign. For the definition of infinite limits, the concept of $\epsilon$-neighbourhoods, $\epsilon>0$, of the symbols $+\infty$, $-\infty$ or $\infty$ in the set $\mathbb R$ of real numbers is introduced by the formulas
$$
U(+\infty,\epsilon) = \left\{ x \in \mathbb R \colon x > \frac{1}{\epsilon}\right\},\\
U(-\infty,\epsilon) = \left\{ x \in \mathbb R \colon x < -\frac{1}{\epsilon}\right\},\\
U(\infty,\epsilon) = \left\{ x \in \mathbb R \colon \lvert x \rvert > \frac{1}{\epsilon}\right\},\\
$$
and the concept of the $\epsilon$-neighbourhood of $\infty$ in the set $\mathbb C$ of complex numbers is introduced by the formula
$$
U(\infty,\epsilon) = \left\{ z \in \mathbb C\colon \lvert z \rvert > \frac 1 \epsilon\right\}.
$$

One writes $\lim_{n\to\infty} x_n = \infty$ ($\lim_{n\to \infty} x_n = +\infty$ or $-\infty$), $x_n\in \mathbb R$, $n=1,2,\dots,$ if for each $\epsilon>0$ there is an index $N$ such that for all indices $n>N$ the membership $x_n\in U(\infty,\epsilon)$ ($x_n\in U(+\infty,\epsilon)$ or $x_n\in U(-\infty,\epsilon)$) is satisfied. The infinite limit of a sequence of complex numbers is defined similarly.

A (finite or infinite) limit of a subsequence of a given sequence is called a subsequential limit of the latter. In the set of subsequential limits of any sequence of real numbers there is always a largest one and a smallest one (finite or infinite). The largest (smallest) subsequential limit of a sequence is called its upper (lower) limit. A sequence has a finite or infinite limit if and only if its upper limit coincides with its lower limit, and then their common value is the limit of the sequence.

Other concepts of a limit, for example, the limit of a function and of Riemann sums, can be expressed in terms of the limit of a sequence. The definition of the limit of a sequence can be generalized to directed (partially ordered) sets.

The limit of a function (mapping).

Let and be topological spaces, , an accumulation point (or a cluster point) of , and let be a mapping from into . A point is called a limit of the mapping at (or, as one says, as approaches ), in symbols,

if for any neighbourhood of in there is a neighbourhood of in , such that for any point the image belongs to : . In other words, if . If is a Hausdorff space, then the mapping can have only one limit at a given point .

In the case where and is an accumulation point of , a limit of the restriction of to is called a limit of on , in symbols,

If , , is an accumulation point of , and exists, then the limit of at also exists on and

If , and the limits

exist, then the limit

also exists.

In considering the limit of a mapping (function) , , as , it may happen that or, on the other hand, . The case is of special interest, because it leads to the concept of a continuous function: If , is a Hausdorff space and , then for the mapping to be continuous at it is necessary and sufficient that

always exists, for any mapping , i.e. any mapping is continuous at all isolated points of its domain. Therefore, the concept of a limit of a mapping, in particular, of continuity, is non-trivial only for limit points of the set being mapped (cf. Limit point of a set). In the classical case of the limit of a function it is usually assumed that , i.e. does not belong to the set on which the limit is taken.

If the space satisfies the first axiom of countability at the point and the space is Hausdorff, then for the existence of the limit of a mapping , , it is necessary and sufficient that for any sequence , such that , the limit exists. If this condition holds, the limit does not depend on the choice of the sequence , and the common value of these limits is the limit of at .

The limit of a sequence of points in a topological space is a special case of the limit of a mapping (function): in this case , the set of natural numbers with the discrete topology, , , and a neighbourhood of in is any subset of the form , where is a natural number.

The concept of the limit of a multiple sequence, i.e. of a sequence the members of which are indexed by integral multi-indices, is also a special case of the limit of a mapping.

An intrinsic criterion for the existence of the limit of a mapping at a given point (called the Cauchy criterion) in the case where the space is first countable at and is a complete metric space, is that the limit exists if and only if for each there is a neighbourhood of in such that for all points and satisfying the condition , the inequality holds. In particular, this criterion is valid if is the set of real or complex numbers.

Some properties of the limit.

If is a metric space, , , and the limit exists, then there is a neighbourhood of such that the image of the intersection of the set being mapped and the neighbourhood under the mapping is a bounded subset of .

If a function , , where is the set of real numbers, has a finite non-zero limit at a point , then there exist a neighbourhood of and a number such that for all points the inequalities

are satisfied.

If is a topological group (in particular, an Abelian group with group operation written additively), , , and , then the limit exists and equals if and only if the function has a limit at which is equal to the identity element of (respectively, the function has a limit at which is equal to zero — such functions are called infinitesimal functions).

If is a linear topological space over a field, , and , then the limit of a linear combination of and at is equal to the same linear combination of their limits at the same point:

If is the set of real or complex numbers, (such functions are called numerical) and , then

if , then

and in this case, by the limit is meant the limit of the restriction of to the intersection of the set being mapped and some neighbourhood of , such that on this intersection the quotient is defined. If , , and the limits and exist, then

If and are obtained from the set of real numbers by completing it either with an infinity without a sign or by two signed infinities and , and , , and , then the definition of the limit of a function as defined above is the classical definition of a finite or infinite limit of a real-valued function of one real variable. Similarly, if the spaces and are obtained by completing the set of complex numbers by the infinity , then the definition of a (finite or infinite) limit of a function of a complex variable is obtained. On the other hand, if the space is obtained by completing (), , by the infinity , then the definition of a finite or infinite limit of a function of several variables as the argument approaches a finite point or infinity is obtained.

For functions defined on subsets of the real line (or, more generally, on ordered sets), the concept of a one-sided limit is defined. Examples of functions having at least one one-sided limit at all limit points of their domain are the real-valued monotone functions: If a function is monotone (cf. Monotone function) on a set of the real line and the point is a limit point of , then it is a limit point of at least one of the sets or . If is a limit point of , then has a limit from the left at , i.e. on . On the other hand, if is a limit point of , then has a limit from the right at i.e. on . If, in addition, for example, is increasing and bounded from above, and is a limit point of , then the limit is finite.

A fundamental general method for finding the limit of a function is the determination of the principal parts of the function in a neighbourhood of the given point, which is often done by means of the Taylor formula. For the calculation of a limit, the l'Hospital rule is often useful.

In spite of the great generality of the concept of a limit of a mapping, it does not include all existing concepts of a limit appearing in contemporary mathematics. For example, the concept of the limit of Riemann sums (cf. Integral sum) is not included in the concept of the limit of a mapping (function). A sufficiently general concept of a limit, including all fundamental cases in a certain sense, is the concept of the limit of a mapping with respect to a filter.

The limit of a filter.

Let be a topological space, , let be a base for its topology, and let be a filter on (i.e. a non-empty family of non-empty subsets of such that for any there is an such that and such that every subset of containing an belongs to ). A point is called a limit of the filter , or its limit point, if is stronger than the filter consisting of a local base for the topology at , i.e. for any there is an such that .

Let be the set of natural numbers with the discrete topology. The filter on consisting of the complements of all finite subsets of is called the natural filter on and is denoted by . It does not have a limit in . The same filter on the set , in which the local base consists of the sets and consists of the singleton for , has as its limit. Uniqueness of the limit of a filter on a topological space is connected with being able to separate points of the space; in order that every filter on a topological space has at most one limit it is necessary and sufficient that the space be a Hausdorff space.

Let be a set, a topological space, a mapping from into , and a filter on . A point is called the limit of the mapping with respect to the filter , in symbols,

if the filter consisting of all sets , , has as its limit in .

If is the set of natural numbers, is a mapping from into a topological space , , , and is the natural filter, then the limit of with respect to in coincides with the usual limit of the sequence in .

If in , the filter on is the product of two natural filters , i.e. it consists of all sets of the form , where , and if is a mapping from into a topological space , , , then the limit of with respect to in coincides with the usual limit of the double sequence in .

Let the elements of a set , in turn, be the sets consisting of a partition of some interval , , and points , i.e.

Let (for any ) be the subset of consisting of all elements for which the mesh of the partition appearing in is smaller than , i.e.

The system is a filter. Every real-valued function defined on induces a mapping of into by the formula

Therefore, is the Riemann sum of corresponding to .

The limit of in with respect to coincides with the usual limit of the Riemann sums of as the mesh of the partition converges to zero. This coincidence holds in the sense that both limits simultaneously exist or not, and if they do so, then they are equal and coincide with the Riemann integral of over .

The limit of a mapping of topological spaces with respect to a filter.

Let and be topological spaces, , let be a filter on , and let be a mapping from into . A point is called the limit of the mapping at with respect to the filter if is the limit of and is the limit of ; notation:

If is a neighbourhood base at , and the filter consists of all "punctured" neighbourhoods of , , then the limit coincides with the usual limit of at , i.e. it generalizes the classical definition of the limit of a mapping as formulated in terms of neighbourhoods. An immediate generalization of the concept of the limit of a sequence is the limit of a directed set in a topological space, i.e. a partially ordered set in which any two elements have a common successor. The concept of a limit of a mapping from one topological space into another can be formulated in terms of limits with respect to directed sets (cf. Generalized sequence; Convergence, types of).

The limit of a sequence of sets.

The topological limit. Let , be subsets of a topological space . The upper topological limit of the sequence is, by definition, the set of those points every neighbourhood of which intersects infinitely many sets . The lower topological limit is the set of those points every neighbourhood of which contains points of almost-all sets . It is obvious that . If , then the sequence is called convergent and the set its topological limit; one writes . The upper and lower topological limits of a sequence are closed sets.

The set-theoretical limit. There is a concept of the limit of a sequence of sets not involving topology. A sequence , of sets is called convergent if there is a set , called its limit and denoted by

such that every element of belongs to all sets from some index onwards and such that every point of the union of all not belonging to is contained in only finitely many sets . The set is the limit of the sequence if and only if the upper and lower limits of the sequence coincide and are equal to .

Comments

For the general theory of topological spaces one needs to study more general convergence concepts than that of sequences of points. One needs to consider limits of sets of points indexed by a directed partially ordered set (cf. Directed set). Such sets are called nets, and the convergence of nets is sometimes known as Moore–Smith convergence.

The basic (inductive) idea of "limit" is that of some object being approximated arbitrarily closely by a sequence of other objects. There are also various algebraic and categorical implementations of this idea.

Let be a category and a partially ordered set. A diagram in indexed by consists of an object for each and a morphism for each in , such that , if .

The projective limit (or inverse limit) of consists of an object together with morphisms such that if and such that if, moreover, and are such that for all , then there is a unique morphism such that . Notation: .

Dually, the inductive limit (or directed limit) of consists of an object together with morphisms such that for all and such that if, moreover, and are such that for all , then there is a unique morphism such that . Notation: .

These are very general notions and include direct products and direct sums (when every pair of points in is incomparable). In case is directed (cf. Directed set), the idea of better and better approximations again reemerges; cf. e.g. Topological vector space.

More generally one also considers projective and inductive limits of diagrams in categories (cf. Diagram).

Still another limit concept is that of the limit of a spectral sequence: If the spectral sequence converges to, say, , then is also called the limit of .